54 lines
1.8 KiB
Mathematica
54 lines
1.8 KiB
Mathematica
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%%*************************************************************************
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%% ToeplitzApprox: find the nearest symmetric positive definite Toeplitz
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%% matrix to a given symmetric matrix F.
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%%
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%% max -y(n+1)
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%% s.t. T(y(1:n)) + y(n+1)*B >= 0
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%% [I 0 ] + sum_{k=1}^n y(k) [0 gam(k)*e_k ] + y(n+1)*B >= 0
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%% [0 -beta] [gam(k)*e_k' -2q(k) ]
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%%
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%% where B = diag([zeros(n,1); 1])
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%% q(1) = - Tr(F); q(k+1) = -sum of upper and lower kth diagonals of F
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%% gam(1) = sqrt(n); gam(k) = sqrt(2*(n-k+1)) for k=2:n
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%% beta = norm(F,'fro')^2
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%%*****************************************************************
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%% SDPT3: version 4.0
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%% Copyright (c) 1997 by
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%% Kim-Chuan Toh, Michael J. Todd, Reha H. Tutuncu
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%% Last Modified: 16 Sep 2004
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%%*****************************************************************
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function [blk,At,C,b] = ToeplitzApprox(F)
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n = length(F);
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gam = sqrt([n, 2*(n-1:-1:1)]);
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q = zeros(n,1);
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q(1) = -sum(diag(F));
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for k=1:n-1
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q(k+1) = -2*sum(diag(F,k));
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end
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beta = norm(F,'fro')^2;
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blk{1,1} = 's'; blk{1,2} = n+1;
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blk{2,1} = 's'; blk{2,2} = n+1;
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b = [zeros(n,1); -1];
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C{1,1} = sparse(n+1,n+1);
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C{2,1} = spdiags([ones(n,1); -beta],0,n+1,n+1);
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Acell = cell(1,n+1);
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Acell{1} = -spdiags([ones(n,1); 0],0,n+1,n+1);
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for k = 1:n-1
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tmp = -spdiags([ones(n,1); 0],k,n+1,n+1);
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Acell{k+1} = tmp + tmp';
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end
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Acell{n+1} = -spconvert([n+1,n+1,1]);
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At(1,1) = svec(blk(1,:),Acell,1);
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for k = 1:n
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Acell{k} = -spconvert([k, n+1, gam(k); n+1, k, gam(k); n+1, n+1, -2*q(k)]);
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end
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Acell{n+1} = -spconvert([n+1,n+1,1]);
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At(2,1) = svec(blk(2,:),Acell,1);
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%%***********************************************************************
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